Statistical Analysis of Karcher Means for Random Restricted PSD Matrices
Machine Learning
2023-03-22 v3 Machine Learning
Abstract
Non-asymptotic statistical analysis is often missing for modern geometry-aware machine learning algorithms due to the possibly intricate non-linear manifold structure. This paper studies an intrinsic mean model on the manifold of restricted positive semi-definite matrices and provides a non-asymptotic statistical analysis of the Karcher mean. We also consider a general extrinsic signal-plus-noise model, under which a deterministic error bound of the Karcher mean is provided. As an application, we show that the distributed principal component analysis algorithm, LRC-dPCA, achieves the same performance as the full sample PCA algorithm. Numerical experiments lend strong support to our theories.
Keywords
Cite
@article{arxiv.2302.12426,
title = {Statistical Analysis of Karcher Means for Random Restricted PSD Matrices},
author = {Hengchao Chen and Xiang Li and Qiang Sun},
journal= {arXiv preprint arXiv:2302.12426},
year = {2023}
}